Noise reduced circuits for trapped-ion quantum computers

ABSTRACT

Embodiments described herein are generally related to a method and a system for performing a computation using a hybrid quantum-classical computing system, and, more specifically, to providing an approximate solution to an optimization problem using a hybrid quantum-classical computing system that includes a group of trapped ions. A hybrid quantum-classical computing system that is able to provide a solution to a combinatorial optimization problem may include a classical computer, a system controller, and a quantum processor. The methods and systems described herein include an efficient and noise resilient method for constructing trial states in the quantum processor in solving a problem in a hybrid quantum-classical computing system, which provides improvement over the conventional method for computation in a hybrid quantum-classical computing system.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit to U.S. Provisional Application No.62/852,264, filed May 23, 2019, which is incorporated by referenceherein.

BACKGROUND Field

The present disclosure generally relates to a method of performingcomputation in a hybrid quantum-classical computing system, and morespecifically, to a method of solving an optimization problem in a hybridcomputing system that includes a classical computer and quantum computerthat includes a group of trapped ions.

Description of the Related Art

In quantum computing, quantum bits or qubits, which are analogous tobits representing a “0” and a “1” in a classical (digital) computer, arerequired to be prepared, manipulated, and measured (read-out) with nearperfect control during a computation process. Imperfect control of thequbits leads to errors that can accumulate over the computation process,limiting the size of a quantum computer that can perform reliablecomputations.

Among physical systems upon which it is proposed to build large-scalequantum computers, is a group of ions (e.g., charged atoms), which aretrapped and suspended in vacuum by electromagnetic fields. The ions haveinternal hyperfine states which are separated by frequencies in theseveral GHz range and can be used as the computational states of a qubit(referred to as “qubit states”). These hyperfine states can becontrolled using radiation provided from a laser, or sometimes referredto herein as the interaction with laser beams. The ions can be cooled tonear their motional ground states using such laser interactions. Theions can also be optically pumped to one of the two hyperfine stateswith high accuracy (preparation of qubits), manipulated between the twohyperfine states (single-qubit gate operations) by laser beams, andtheir internal hyperfine states detected by fluorescence uponapplication of a resonant laser beam (read-out of qubits). A pair ofions can be controllably entangled (two-qubit gate operations) byqubit-state dependent force using laser pulses that couple the ions tothe collective motional modes of a group of trapped ions, which arisefrom their Coulombic interaction between the ions. In general,entanglement occurs when pairs or groups of ions (or particles) aregenerated, interact, or share spatial proximity in ways such that thequantum state of each ion cannot be described independently of thequantum state of the others, even when the ions are separated by a largedistance.

In current state-of-the-art quantum computers, control of qubits isimperfect (noisy) and the number of qubits used in these quantumcomputers generally range from a hundred qubits to thousands of qubits.The number of quantum gates that can be used in such a quantum computer(referred to as a “noisy intermediate-scale quantum device” or “NISQdevice”) to construct circuits to run an algorithm within a controllederror rate is limited due to the noise.

For solving some optimization problems, a NISQ device having shallowcircuits (with small number of gate operations to be executed intime-sequence) can be used in combination with a classical computer(referred to as a hybrid quantum-classical computing system). Inparticular, finding low-energy states of a many-particle quantum system,such as large molecules, or in finding an approximate solution tocombinatorial optimization problems, a quantum subroutine, which is runon a NISQ device, can be run as part of a classical optimizationroutine, which is run on a classical computer. The classical computer(also referred to as a “classical optimizer”) instructs a controller toprepare the NISQ device (also referred to as a “quantum processor”) inan N-qubit state, execute quantum gate operations, and measure anoutcome of the quantum processor. Subsequently, the classical optimizerinstructs the controller to prepare the quantum processor in a slightlydifferent N-qubit state, and repeats execution of the gate operation andmeasurement of the outcome. This cycle is repeated until the approximatesolution can be extracted. Such hybrid quantum-classical computingsystem having an NISQ device may outperform classical computers infinding low-energy states of a many-particle quantum system and infinding approximate solutions to such combinatorial optimizationproblems. However, the number of quantum gate operations required withinthe quantum routine increases rapidly as the problem size increases,leading to accumulated errors in the NISQ device and causing theoutcomes of these processes to be not reliable.

Therefore, there is a need for a procedure to construct shallow circuitsthat require a minimum number of quantum gate operations to performcomputation and thus reduce noise in a hybrid quantum-classicalcomputing system.

SUMMARY

A method of performing computation in a hybrid quantum-classicalcomputing system includes computing, by a classical computer, a modelHamiltonian including a plurality of sub-Hamiltonian onto which aselected problem is mapped, setting a quantum processor in an initialstate, where the quantum processor comprises a plurality of trappedions, each of which has two frequency-separated states defining a qubit,transforming the quantum processor from the initial state to a trialstate based on each of the plurality of sub-Hamiltonians and an initialset of variational parameters by applying a reduced trial statepreparation circuit to the quantum processor, measuring an expectationvalue of each of the plurality of sub-Hamiltonians on the quantumprocessor, and determining, by the classical computer, if a differencebetween the measured expectation value of the model Hamiltonian is moreor less than a predetermined value. If it is determined that thedifference is more than the predetermined value, the classical computereither selects another set of variational parameters based on aclassical optimization method, sets the quantum processor in the initialstate, transforms the quantum processor from the initial state to a newtrial state based on each of the plurality of sub-Hamiltonians and theanother set of variational parameters by applying a new reduced trialstate preparation circuit to the quantum processor, and measures anexpectation value of the each of the plurality of sub-Hamiltonians onthe quantum processor after transforming the quantum processor to thenew trial state. If it is determined that the difference is less thanthe predetermined value, the classical computer outputs the measuredexpectation value of the model Hamiltonian as an optimized solution tothe selected problem.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above-recited features of the presentdisclosure can be understood in detail, a more particular description ofthe disclosure, briefly summarized above, may be had by reference toembodiments, some of which are illustrated in the appended drawings. Itis to be noted, however, that the appended drawings illustrate onlytypical embodiments of this disclosure and are therefore not to beconsidered limiting of its scope, for the disclosure may admit to otherequally effective embodiments.

FIG. 1 is a schematic partial view of an ion trap quantum computingsystem according to one embodiment.

FIG. 2 depicts a schematic view of an ion trap for confining ions in agroup according to one embodiment.

FIG. 3 depicts a schematic energy diagram of each ion in a group oftrapped ions according to one embodiment.

FIG. 4 depicts a qubit state of an ion represented as a point on asurface of the Bloch sphere.

FIGS. 5A, 5B, and 5C depict a few schematic collective transversemotional mode structures of a group of five trapped ions.

FIGS. 6A and 6B depict schematic views of motional sideband spectrum ofeach ion and a motional mode according to one embodiment.

FIG. 7 depicts an overall hybrid quantum-classical computing system forobtaining a solution to an optimization problem by Variational QuantumEigensolver (VQE) algorithm or Quantum Approximate OptimizationAlgorithm (QAOA) according to one embodiment.

FIG. 8 depicts a flowchart illustrating a method of obtaining a solutionto an optimization problem by Variational Quantum Eigensolver (VQE)algorithm or Quantum Approximate Optimization Algorithm (QAOA) accordingto one embodiment.

FIG. 9A illustrates a trial state preparation circuit according to oneembodiment.

FIGS. 9B and 9C illustrate reduced trial state preparation circuitsaccording to one embodiment.

To facilitate understanding, identical reference numerals have beenused, where possible, to designate identical elements that are common tothe figures. In the figures and the following description, an orthogonalcoordinate system including an X-axis, a Y-axis, and a Z-axis is used.The directions represented by the arrows in the drawing are assumed tobe positive directions for convenience. It is contemplated that elementsdisclosed in some embodiments may be beneficially utilized on otherimplementations without specific recitation.

DETAILED DESCRIPTION

Embodiments described herein are generally related to a method and asystem for performing a computation using a hybrid quantum-classicalcomputing system, and, more specifically, to providing an approximatesolution to an optimization problem using a hybrid quantum-classicalcomputing system that includes a group of trapped ions.

A hybrid quantum-classical computing system that is able to provide asolution to a combinatorial optimization problem may include a classicalcomputer, a system controller, and a quantum processor. In someembodiments, the system controller is housed within the classicalcomputer. The classical computer performs supporting and system controltasks including selecting a combinatorial optimization problem to be runby use of a user interface, running a classical optimization routine,translating the series of logic gates into laser pulses to apply on thequantum processor, and pre-calculating parameters that optimize thelaser pulses by use of a central processing unit (CPU). A softwareprogram for performing the tasks is stored in a non-volatile memorywithin the classical computer.

The quantum processor includes trapped ions that are coupled withvarious hardware, including lasers to manipulate internal hyperfinestates (qubit states) of the trapped ions and an acousto-optic modulatorto read-out the internal hyperfine states (qubit states) of the trappedions. The system controller receives from the classical computerinstructions for controlling the quantum processor, controls varioushardware associated with controlling any and all aspects used to run theinstructions for controlling the quantum processor, and returns aread-out of the quantum processor and thus output of results of thecomputation(s) to the classical computer.

The methods and systems described herein include an efficient method forconstructing quantum gate operations executed by the quantum processorin solving a problem in a hybrid quantum-classical computing system.

General Hardware Configurations

FIG. 1 is a schematic partial view of an ion trap quantum computingsystem, or system 100, according to one embodiment. The system 100includes a classical (digital) computer 102, a system controller 104 anda quantum processor that is a group 106 of trapped ions (i.e., fiveshown) that extend along the Z-axis. The classical computer 102 includesa central processing unit (CPU), memory, and support circuits (or I/O).The memory is connected to the CPU, and may be one or more of a readilyavailable memory, such as a read-only memory (ROM), a random accessmemory (RAM), floppy disk, hard disk, or any other form of digitalstorage, local or remote. Software instructions, algorithms and data canbe coded and stored within the memory for instructing the CPU. Thesupport circuits (not shown) are also connected to the CPU forsupporting the processor in a conventional manner. The support circuitsmay include conventional cache, power supplies, clock circuits,input/output circuitry, subsystems, and the like.

An imaging objective 108, such as an objective lens with a numericalaperture (NA), for example, of 0.37, collects fluorescence along theY-axis from the ions and maps each ion onto a multi-channelphoto-multiplier tube (PMT) 110 for measurement of individual ions.Non-copropagating Raman laser beams from a laser 112, which are providedalong the X-axis, perform operations on the ions. A diffractive beamsplitter 114 creates an array of static Raman beams 116 that areindividually switched using a multi-channel acousto-optic modulator(AOM) 118 and is configured to selectively act on individual ions. Aglobal Raman laser beam 120 illuminates all ions at once. The systemcontroller (also referred to as a “RF controller”) 104 controls the AOM118 and thus controls laser pulses to be applied to trapped ions in thegroup 106 of trapped ions. The system controller 104 includes a centralprocessing unit (CPU) 122, a read-only memory (ROM) 124, a random accessmemory (RAM) 126, a storage unit 128, and the like. The CPU 122 is aprocessor of the system controller 104. The ROM 124 stores variousprograms and the RAM 126 is the working memory for various programs anddata. The storage unit 128 includes a nonvolatile memory, such as a harddisk drive (HDD) or a flash memory, and stores various programs even ifpower is turned off. The CPU 122, the ROM 124, the RAM 126, and thestorage unit 128 are interconnected via a bus 130. The system controller104 executes a control program which is stored in the ROM 124 or thestorage unit 128 and uses the RAM 126 as a working area. The controlprogram will include software applications that include program codethat may be executed by processor in order to perform variousfunctionalities associated with receiving and analyzing data andcontrolling any and all aspects of the methods and hardware used tocreate the ion trap quantum computer system 100 discussed herein.

FIG. 2 depicts a schematic view of an ion trap 200 (also referred to asa Paul trap) for confining ions in the group 106 according to oneembodiment. The confining potential is exerted by both static (DC)voltage and radio frequency (RF) voltages. A static (DC) voltage V_(S)is applied to end-cap electrodes 210 and 212 to confine the ions alongthe Z-axis (also referred to as an “axial direction” or a “longitudinaldirection”). The ions in the group 106 are nearly evenly distributed inthe axial direction due to the Coulomb interaction between the ions. Insome embodiments, the ion trap 200 includes four hyperbolically-shapedelectrodes 202, 204, 206, and 208 extending along the Z-axis.

During operation, a sinusoidal voltage V₁ (with an amplitude V_(RF)/2)is applied to an opposing pair of the electrodes 202, 204 and asinusoidal voltage V₂ with a phase shift of 180° from the sinusoidalvoltage V₁ (and the amplitude V_(RF)/2) is applied to the other opposingpair of the electrodes 206, 208 at a driving frequency ω_(RF),generating a quadrupole potential. In some embodiments, a sinusoidalvoltage is only applied to one opposing pair of the electrodes 202, 204,and the other opposing pair 206, 208 is grounded. The quadrupolepotential creates an effective confining force in the X-Y planeperpendicular to the Z-axis (also referred to as a “radial direction” or“transverse direction”) for each of the trapped ions, which isproportional to a distance from a saddle point (i.e., a position in theaxial direction (Z-direction)) at which the RF electric field vanishes.The motion in the radial direction (i.e., direction in the X-Y plane) ofeach ion is approximated as a harmonic oscillation (referred to assecular motion) with a restoring force towards the saddle point in theradial direction and can be modeled by spring constants k_(x) and k_(y),respectively, as is discussed in greater detail below. In someembodiments, the spring constants in the radial direction are modeled asequal when the quadrupole potential is symmetric in the radialdirection. However, undesirably in some cases, the motion of the ions inthe radial direction may be distorted due to some asymmetry in thephysical trap configuration, a small DC patch potential due toinhomogeneity of a surface of the electrodes, or the like and due tothese and other external sources of distortion the ions may lieoff-center from the saddle points.

FIG. 3 depicts a schematic energy diagram 300 of each ion in the group106 of trapped ions according to one embodiment. In one example, eachion may be a positive Ytterbium ion, ¹⁷¹Yb⁺, which has the ²S_(1/2)hyperfine states (i.e., two electronic states) with an energy splitcorresponding to a frequency difference (referred to as a “carrierfrequency”) of ω₀₁/2π=12.642821 GHz. A qubit is formed with the twohyperfine states, denoted as |0

and |1

, where the hyperfine ground state (i.e., the lower energy state of the²S_(1/2) hyperfine states) is chosen to represent |0

. Hereinafter, the terms “hyperfine states,” “internal hyperfinestates,” and “qubits” may be interchangeably used to represent |0

and |1

. Each ion may be cooled (i.e., kinetic energy of the ion may bereduced) to near the motional ground state |0

_(m) for any motional mode m with no phonon excitation (i.e., n_(ph)=0)by known laser cooling methods, such as Doppler cooling or resolvedsideband cooling, and then the qubit state prepared in the hyperfineground state |0

by optical pumping. Here, |0

represents the individual qubit state of a trapped ion whereas |0

_(m) with the subscript m denotes the motional ground state for amotional mode m of a group 106 of trapped ions.

An individual qubit state of each trapped ion may be manipulated by, forexample, a mode-locked laser at 355 nanometers (nm) via the excited²P_(1/2) level (denoted as |e

). As shown in FIG. 3, a laser beam from the laser may be split into apair of non-copropagating laser beams (a first laser beam with frequencyω₁ and a second laser beam with frequency ω₂) in the Ramanconfiguration, and detuned by a one-photon transition detuning frequencyΔ=ω₁−ω_(0e) with respect to the transition frequency ω_(0e) between |0

and |e

, as illustrated in FIG. 3. A two-photon transition detuning frequency δincludes adjusting the amount of energy that is provided to the trappedion by the first and second laser beams, which when combined is used tocause the trapped ion to transfer between the hyperfine states |0

and |1

. When the one-photon transition detuning frequency Δ is much largerthan a two-photon transition detuning frequency (also referred to simplyas “detuning frequency”) δ=ω₁−ω₂−ω₀₁ (hereinafter denoted as ±μ, μ beinga positive value), single-photon Rabi frequencies Ω_(0e)(t) andΩ_(1e)(t) (which are time-dependent, and are determined by amplitudesand phases of the first and second laser beams), at which Rabi floppingbetween states |0

and |e

and between states |1

and |e

respectively occur, and a spontaneous emission rate from the excitedstate |e

, Rabi flopping between the two hyperfine states |0

and |1

(referred to as a “carrier transition”) is induced at the two-photonRabi frequency Ω(t). The two-photon Rabi frequency Ω(t) has an intensity(i.e., absolute value of amplitude) that is proportional toΩ_(0e)Ω_(1e)/2Δ, where Ω_(0e) and Ω_(1e) are the single-photon Rabifrequencies due to the first and second laser beams, respectively.Hereinafter, this set of non-copropagating laser beams in the Ramanconfiguration to manipulate internal hyperfine states of qubits (qubitstates) may be referred to as a “composite pulse” or simply as a“pulse,” and the resulting time-dependent pattern of the two-photon Rabifrequency Ω(t) may be referred to as an “amplitude” of a pulse or simplyas a “pulse,” which are illustrated and further described below. Thedetuning frequency δ=ω₁−ω₂−ω₀₁ may be referred to as detuning frequencyof the composite pulse or detuning frequency of the pulse. The amplitudeof the two-photon Rabi frequency Ω(t), which is determined by amplitudesof the first and second laser beams, may be referred to as an“amplitude” of the composite pulse.

It should be noted that the particular atomic species used in thediscussion provided herein is just one example of atomic species whichhas stable and well-defined two-level energy structures when ionized andan excited state that is optically accessible, and thus is not intendedto limit the possible configurations, specifications, or the like of anion trap quantum computer according to the present disclosure. Forexample, other ion species include alkaline earth metal ions (Be⁺, Ca⁺,Sr⁺, Mg+, and Ba⁺) or transition metal ions (Zn⁺, Hg⁺, Cd⁺).

FIG. 4 is provided to help visualize a qubit state of an ion isrepresented as a point on a surface of the Bloch sphere 400 with anazimuthal angle ϕ and a polar angle θ. Application of the compositepulse as described above, causes Rabi flopping between the qubit state|0

(represented as the north pole of the Bloch sphere) and |1

(the south pole of the Bloch sphere) to occur. Adjusting time durationand amplitudes of the composite pulse flips the qubit state from |0

to |1

(i.e., from the north pole to the south pole of the Bloch sphere), orthe qubit state from |1

to |0

(i.e., from the south pole to the north pole of the Bloch sphere). Thisapplication of the composite pulse is referred to as a “π-pulse”.Further, by adjusting time duration and amplitudes of the compositepulse, the qubit state |0

may be transformed to a superposition state |0

+|1

, where the two qubit states |0

and |1

are added and equally-weighted in-phase (a normalization factor of thesuperposition state is omitted hereinafter without loss of generality)and the qubit state |1

to a superposition state |0

−|1

, where the two qubit states |0

and |1

are added equally-weighted but out of phase. This application of thecomposite pulse is referred to as a “π/2-pulse”. More generally, asuperposition of the two qubits states |0

and |1

that are added and equally-weighted is represented by a point that lieson the equator of the Bloch sphere. For example, the superpositionstates |0

±|1

correspond to points on the equator with the azimuthal angle ϕ beingzero and π, respectively. The superposition states that correspond topoints on the equator with the azimuthal angle ϕ are denoted as |0

+e^(iϕ)|1

(e.g., |0

±i|1

for ϕ=±π/2). Transformation between two points on the equator (i.e., arotation about the Z-axis on the Bloch sphere) can be implemented byshifting phases of the composite pulse.

Entanglement Formation

FIGS. 5A, 5B, and 5C depict a few schematic structures of collectivetransverse motional modes (also referred to simply as “motional modestructures”) of a group 106 of five trapped ions, for example. Here, theconfining potential due to a static voltage V_(S) applied to the end-capelectrodes 210 and 212 is weaker compared to the confining potential inthe radial direction. The collective motional modes of the group 106 oftrapped ions in the transverse direction are determined by the Coulombinteraction between the trapped ions combined with the confiningpotentials generated by the ion trap 200. The trapped ions undergocollective transversal motions (referred to as “collective transversemotional modes,” “collective motional modes,” or simply “motionalmodes”), where each mode has a distinct energy (or equivalently, afrequency) associated with it. A motional mode having the m-th lowestenergy is hereinafter referred to as |n_(ph)

_(m), where n_(ph) denotes the number of motional quanta (in units ofenergy excitation, referred to as phonons) in the motional mode, and thenumber of motional modes M in a given transverse direction is equal tothe number of trapped ions N in the group 106. FIGS. 5A-5C schematicallyillustrates examples of different types of collective transversemotional modes that may be experienced by five trapped ions that arepositioned in a group 106. FIG. 5A is a schematic view of a commonmotional mode |n_(ph)

_(M) having the highest energy, where M is the number of motional modes.In the common motional mode |n

_(M), all ions oscillate in phase in the transverse direction. FIG. 5Bis a schematic view of a tilt motional mode |n_(ph)

_(M-1) which has the second highest energy. In the tilt motional mode,ions on opposite ends move out of phase in the transverse direction(i.e., in opposite directions). FIG. 5C is a schematic view of ahigher-order motional mode |n_(ph)

_(M-3) which has a lower energy than that of the tilt motional mode|n_(ph)

_(M-1), and in which the ions move in a more complicated mode pattern.

It should be noted that the particular configuration described above isjust one among several possible examples of a trap for confining ionsaccording to the present disclosure and does not limit the possibleconfigurations, specifications, or the like of traps according to thepresent disclosure. For example, the geometry of the electrodes is notlimited to the hyperbolic electrodes described above. In other examples,a trap that generates an effective electric field causing the motion ofthe ions in the radial direction as harmonic oscillations may be amulti-layer trap in which several electrode layers are stacked and an RFvoltage is applied to two diagonally opposite electrodes, or a surfacetrap in which all electrodes are located in a single plane on a chip.Furthermore, a trap may be divided into multiple segments, adjacentpairs of which may be linked by shuttling one or more ions, or coupledby photon interconnects. A trap may also be an array of individualtrapping regions arranged closely to each other on a micro-fabricatedion trap chip. In some embodiments, the quadrupole potential has aspatially varying DC component in addition to the RF component describedabove.

In an ion trap quantum computer, the motional modes may act as a databus to mediate entanglement between two qubits and this entanglement isused to perform an XX gate operation. That is, each of the two qubits isentangled with the motional modes, and then the entanglement istransferred to an entanglement between the two qubits by using motionalsideband excitations, as described below. FIGS. 6A and 6B schematicallydepict views of a motional sideband spectrum for an ion in the group 106in a motional mode |n_(ph)

_(M) having frequency ω_(m) according to one embodiment. As illustratedin FIG. 6B, when the detuning frequency of the composite pulse is zero(i.e., a frequency difference between the first and second laser beamsis tuned to the carrier frequency, δ=ω₁−ω₂−ω₀₁=0), simple Rabi floppingbetween the qubit states |0

and |1

(carrier transition) occurs. When the detuning frequency of thecomposite pulse is positive (i.e., the frequency difference between thefirst and second laser beams is tuned higher than the carrier frequency,δ=ω₁−ω₂−ω₀₁=μ>0, referred to as a blue sideband), Rabi flopping betweencombined qubit-motional states |0

|n_(ph)

_(m) and |1

|n_(ph)+1

_(m) occurs (i.e., a transition from the m-th motional mode withn-phonon excitations denoted by |n_(ph)

_(m) to the m-th motional mode with (n_(ph)+1)-phonon excitationsdenoted by |n_(ph)+1

_(m) occurs when the qubit state |0

flips to |1

). When the detuning frequency of the composite pulse is negative (i.e.,the frequency difference between the first and second laser beams istuned lower than the carrier frequency by the frequency ω_(m) of themotional mode |n_(ph)

_(m), δ=ω₁−ω₂−ω₀₁=−μ<0, referred to as a red sideband), Rabi floppingbetween combined qubit-motional states |0

|n_(ph)

_(m) and |1

|n_(ph)−1

_(m) occurs (i.e., a transition from the motional mode |n_(ph)

_(m) to the motional mode |n_(ph)−1

_(m) with one less phonon excitations occurs when the qubit state |0

flips to |1

). A π/2-pulse on the blue sideband applied to a qubit transforms thecombined qubit-motional state |0

|n_(ph)

_(m) into a superposition of |0

|n_(ph)

_(m) and |1

|n_(ph)+1

_(m). A π/2-pulse on the red sideband applied to a qubit transforms thecombined qubit-motional |0

|n_(ph)

_(m) into a superposition of |0

|n_(ph)

_(m) and |1

|n_(ph)−1

_(m). When the two-photon Rabi frequency Ω(t) is smaller as compared tothe detuning frequency δ=ω₁−ω₂−ω₀₁=±μ, the blue sideband transition orthe red sideband transition may be selectively driven. Thus, qubitstates of a qubit can be entangled with a desired motional mode byapplying the right type of pulse, such as a π/2-pulse, which can besubsequently entangled with another qubit, leading to an entanglementbetween the two qubits that is needed to perform an XX-gate operation inan ion trap quantum computer.

By controlling and/or directing transformations of the combinedqubit-motional states as described above, an XX-gate operation may beperformed on two qubits (i-th and j-th qubits). In general, the XX-gateoperation (with maximal entanglement) respectively transforms two-qubitstates |0

_(i)|0

_(j), |0

_(i)|1

_(j), |1

_(i)|0

_(j), and as follows:|0

_(i)|0

_(j)→|0

_(i)|0

_(j) −i|1

_(i)|1

_(j)|0

_(i)|1

_(j)→|0

_(i)|1

_(j) −i|1

_(i)|0

_(j)|1

_(i)|0

_(j) →−i|0

_(i)|1

_(j)+|1

_(i)|0

_(j)|1

_(i)|1

_(j) →−i|0

_(i)|1

_(j)+|1

_(i)|1

_(j).For example, when the two qubits (i-th and j-th qubits) are bothinitially in the hyperfine ground state |0

(denoted as |0

_(i)|0

_(j)) and subsequently a π/2-pulse on the blue sideband is applied tothe i-th qubit, the combined state of the i-th qubit and the motionalmode |0

_(i)|n_(ph)

_(m) is transformed into a superposition of |0

_(i)|n_(ph)

_(m) and |1

_(i)|n_(ph)+1

_(m), and thus the combined state of the two qubits and the motionalmode is transformed into a superposition of |0

_(i)|0

_(j)|n_(ph)

_(m) and |1

_(i)|0

_(j)|n_(ph)+1

_(m). When a π/2-pulse on the red sideband is applied to the j-th qubit,the combined state of the j-th qubit and the motional mode |0

_(j)|n_(ph)

_(m) is transformed to a superposition of |0

_(j)|n_(ph)

_(m) and |1

_(j)|n_(ph)−1

_(m) and the combined state |0

_(j)|n_(ph)+1

_(m) is transformed into a superposition of |0

_(j)|n_(ph)+1

_(m) and.

Thus, applications of a π/2-pulse on the blue sideband on the i-th qubitand a π/2-pulse on the red sideband on the j-th qubit may transform thecombined state of the two qubits and the motional mode |0

_(i)|0

_(j)|n_(ph)

_(m) into a superposition of |0

_(i)|0

_(j)|n_(ph)

_(m) and |1

_(i)|1

_(j)|n_(ph)

_(m), the two qubits now being in an entangled state. For those ofordinary skill in the art, it should be clear that two-qubit states thatare entangled with motional mode having a different number of phononexcitations from the initial number of phonon excitations n_(ph) (i.e.,|1

_(i)|0

_(j)|n_(ph)+1

_(m) and |0

_(i)|1

_(j)|n_(ph)−1

_(m)) can be removed by a sufficiently complex pulse sequence, and thusthe combined state of the two qubits and the motional mode after theXX-gate operation may be considered disentangled as the initial numberof phonon excitations n_(ph) in the m-th motional mode stays unchangedat the end of the XX-gate operation. Thus, qubit states before and afterthe XX-gate operation will be described below generally withoutincluding the motional modes.

More generally, the combined state of i-th and j-th qubits transformedby the application of pulses on the sidebands for duration τ (referredto as a “gate duration”), having amplitudes Ω^((i)) and Ω^((j)) anddetuning frequency μ, can be described in terms of an entanglinginteraction χ^((i,j))(τ) as follows:|0

_(i)|0

_(j)→cos(2χ^((i,j))(τ))|0

_(i)|0

_(j) −i sin(2χ^((i,j)))(τ))|1

_(i)|1

_(j)|0

_(i)|1

_(j)→cos(2χ^((i,j))(τ))|0

_(i)|1

_(j) −i sin(2χ^((i,j)))(τ))|1

_(i)|0

_(j)|1

_(i)|0

_(j) →−i sin(2χ^((i,j)))(τ))|0

_(i)|1

_(nj)+cos(2χ^((i,j))(τ))|1

_(i)|0

_(j)|1

_(i)|1

_(j) →−i sin(2χ^((i,j)))(τ))|0

_(i)|0

_(j)+cos(2χ^((i,j))(τ))|1

_(i)|1

_(j)where,

${\chi^{({i,j})}(\tau)} = {{- 4}{\sum\limits_{m = 1}^{M}{\eta_{m}^{(i)}\eta_{m}^{(j)}{\int\limits_{0}^{\tau}{{dt}_{2}{\int\limits_{0}^{t_{2}}{{dt}_{1}{\Omega^{(i)}\left( t_{2} \right)}{\Omega^{(j)}\left( t_{1} \right)}{\cos\left( {\mu t_{2}} \right)}{\cos\left( {\mu t_{1}} \right)}{\sin\left\lbrack {\omega_{m}\left( {t_{2} - t_{1}} \right)} \right\rbrack}}}}}}}}$and η_(m) ^((i)) is the Lamb-Dicke parameter that quantifies thecoupling strength between the i-th ion and the m-th motional mode havingthe frequency ω_(m), and M is the number of the motional modes (equal tothe number N of ions in the group 106).

The entanglement interaction between two qubits described above can beused to perform an XX-gate operation. The XX-gate operation (XX gate)along with single-qubit operations (R gates) forms a set of gates {R,XX} that can be used to build a quantum computer that is configured toperform desired computational processes. Among several known sets oflogic gates by which any quantum algorithm can be decomposed, a set oflogic gates, commonly denoted as {R, XX}, is native to a quantumcomputing system of trapped ions described herein. Here, the R gatecorresponds to manipulation of individual qubit states of trapped ions,and the XX gate (also referred to as an “entangling gate”) correspondsto manipulation of the entanglement of two trapped ions.

To perform an XX-gate operation between i-th and j-th qubits, pulsesthat satisfy the condition χ^((i,j))(τ)=θ^((i,j))(0<θ^((i,j))≤π/8)(i.e., the entangling interaction χ^((i,j))(τ) has a desired valueθ^((i,j)), referred to as condition for a non-zero entanglementinteraction) are constructed and applied to the i-th and the j-thqubits. The transformations of the combined state of the i-th and thej-th qubits described above corresponds to the XX-gate operation withmaximal entanglement when θ^((i,j))=π/8. Amplitudes Ω^((i))(t) andΩ^((j))(t) of the pulses to be applied to the i-th and the j-th qubitsare control parameters that can be adjusted to ensure a non-zero tunableentanglement of the i-th and the j-th qubits to perform a desired XXgate operation on i-th and j-th qubits.

Hybrid Quantum-Classical Computing System

While currently available quantum computers may be noisy and prone toerrors, a combination of both quantum and classical computers, in whicha quantum computer is a domain-specific accelerator, may be able tosolve optimization problems that are beyond the reach of classicalcomputers. An example of such optimization problems is quantumchemistry, where Variational Quantum Eigensolver (VQE) algorithmsperform a search for the lowest energy (or an energy closest to thelowest energy) of a many-particle quantum system, such as a largemolecules chemical compound and the corresponding state (e.g. aconfiguration of the interacting electrons or spins) by iteratingcomputations between a quantum processor and a classical computer. Amany-particle quantum system in quantum theory is described by a modelHamiltonian and the energy of the many-particle quantum systemcorresponds to the expectation value of the model Hamiltonian. In suchalgorithms, a configuration of electrons or spins that is best knownapproximation calculated by the classical computer is input to thequantum processor as a trial state and the energy of the trial state isestimated using the quantum processor. The classical computer receivesthis estimate, modifies the trial state by a known classicaloptimization algorithm, and returns the modified trial state back to thequantum processor. This iteration is repeated until the estimatereceived from the quantum processor is within a predetermined accuracy.A trial function (i.e., a possible configuration of electrons or spinsof the many-particle quantum system) would require exponentially largeresource to represent on a classical computer, as the number ofelectrons or spins of the many-particle quantum system of interest, butonly require linearly-increasing resource on a quantum processor. Thus,the quantum processor acts as an accelerator for the energy estimationsub-routine of the computation. By solving for a configuration ofelectrons or spins having the lowest energy under differentconfigurations and constraints, a range of molecular reactions can beexplored as part of the solution to this type of optimization problemfor example.

Another example optimization problem is in solving combinatorialoptimization problems, where Quantum Approximate Optimization Algorithm(QAOA) perform search for optimal solutions from a set of possiblesolutions according to some given criteria, using a quantum computer anda classical computer. The combinatorial optimization problems that canbe solved by the methods described herein may include the PageRank (PR)problem for ranking web pages in search engine results and themaximum-cut (MaxCut) problem with applications in clustering, networkscience, and statistical physics. The MaxCut problem aims at groupingnodes of a graph into two partitions by cutting across links betweenthem in such a way that a weighted sum of intersected edges ismaximized. The combinatorial optimization problems that can be solved bythe methods described herein may further include the travelling salesmanproblem for finding shortest and/or cheapest round trips visiting allgiven cities. The travelling salesman problem is applied to scheduling aprinting press for a periodical with multi-editions, scheduling schoolbuses minimizing the number of routes and total distance while no bus isoverloaded or exceeds a maximum allowed policy, scheduling a crew ofmessengers to pick up deposit from branch banks and return the depositto a central bank, determining an optimal path for each army planner toaccomplish the goals of the mission in minimum possible time, designingglobal navigation satellite system (GNSS) surveying networks, and thelike. Another combinatorial optimization problem is the knapsack problemto find a way to pack a knapsack to get the maximum total value, givensome items. The knapsack problem is applied to resource allocation givenfinancial constraints in home energy management, network selection formobile nodes, cognitive radio networks, sensor selection in distributedmultiple radar, or the like.

A combinatorial optimization problem is modeled by an objective function(also referred to as a cost function) that maps events or values of oneor more variables onto real numbers representing “cost” associated withthe events or values and seeks to minimize the cost function. In somecases, the combinatorial optimization problem may seek to maximize theobjective function. The combinatorial optimization problem is furthermapped onto a simple physical system described by a model Hamiltonian(corresponding to the sum of kinetic energy and potential energy of allparticles in the system) and the problem seeks the low-lying energystate of the physical system, as in the case of the Variational QuantumEigensolver (VQE) algorithm.

This hybrid quantum-classical computing system has at least thefollowing advantages. First, an initial guess is derived from aclassical computer, and thus the initial guess does not need to beconstructed in a quantum processor that may not be reliable due toinherent and unwanted noise in the system. Second, a quantum processorperforms a small-sized (e.g., between a hundred qubits an a few thousandqubits) but accelerated operation (that can be performed using a smallnumber of quantum logic gates) between an input of a guess from theclassical computer and a measurement of a resulting state, and thus aNISQ device can execute the operation without accumulating errors. Thus,the hybrid quantum-classical computing system may allow challengingproblems to be solved, such as small but challenging combinatorialoptimization problems, which are not practically feasible on classicalcomputers, or suggest ways to speed up the computation with respect tothe results that would be achieved using the best known classicalalgorithm.

FIGS. 7 and 8 depict an overall hybrid quantum-classical computingsystem 700 and a flowchart illustrating a method 800 of obtaining asolution to an optimization problem by Variational Quantum Eigensolver(VQE) algorithm or Quantum Approximate Optimization Algorithm (QAOA)according to one embodiment. In this example, the quantum processor isthe group 106 of N trapped ions, in which the two hyperfine states ofeach of the N trapped ions form a qubit.

The VQE algorithm relies on a variational search by the well-knownRayleigh-Ritz variational principle. This principle can be used both forsolving quantum chemistry problems by the VQE algorithm andcombinatorial optimization problems solved by the QAOA. The variationalmethod consists of iterations that include choosing a “trial state” ofthe quantum processor depending on a set of one or more parameters(referred to as “variational parameters”) and measuring an expectationvalue of the model Hamiltonian (e.g., energy) of the trial state. A setof variational parameters (and thus a corresponding trial state) isadjusted and an optimal set of variational parameters are found thatminimizes the expectation value of the model Hamiltonian (the energy).The resulting energy is an approximation to the exact lowest energystate. As the processes for obtaining a solution to an optimizationproblem by the VQE algorithm and by the QAOA, the both processes aredescribed in parallel below

In block 802, by the classical computer 102, an optimization problem tobe solved by the VQE algorithm or the QAOA is selected, for example, byuse of a user interface of the classical computer 102, or retrieved fromthe memory of the classical computer 102, and a model Hamiltonian H_(C),which describes a many-particle quantum system in the quantum chemistryproblem, or to which the selected combinatorial optimization problem ismapped, is computed.

In a quantum chemistry problem defined on an N-spin system, the systemcan be well described by a model Hamiltonian that includes quantum spins(each denoted by the third Pauli matrix σ_(i) ^(z)) (i=1, 2, . . . , N)and couplings among the quantum spins σ_(i) ^(z), H_(C)=Σ_(α=1)^(t)h_(α)P_(α), where P_(α) is a Pauli string (also referred to as aPauli term) P_(α)=σ₁ ^(α) ¹ ⊗σ₂ ^(α) ² ⊗ . . . σ_(N) ^(α) ^(N) and σ_(i)^(α) ^(i) is either the identity operator I or the Pauli matrix σ_(i)^(X), σ_(i) ^(Y), or σ_(i) ^(Z). Here t stands for the number ofcouplings among the quantum spins and h_(α) (α=1, 2, . . . , t) standsfor the strength of the coupling α. An N-electron system can be alsodescribed by the same model Hamiltonian H_(C)=Σ_(α=1) ^(t)h_(α)P_(α).The goal is to find low-lying energy states of the model HamiltonianH_(C).

In a combinatorial optimization problem defined on a set of N binaryvariables with t constrains (α=1, 2, . . . t), the objective function isthe number of satisfied clauses C(z)=Σ_(α=1) ^(t)C_(α)(z) or a weightedsum of satisfied clauses C(z)=Σ_(α=1) ^(t)h_(α)C_(α)(z) (h_(α)corresponds to a weight for each constraint α), where z=z₁ z₂ . . .z_(N) is a N-bit string and C_(α)(z)=1 if z satisfies the constraint α.The clause C_(α)(z) that describes the constraint α typically includes asmall number of variables z_(i). The goal is to minimize the objectivefunction. Minimizing this objective function can be converted to findinga low-lying energy state of a model Hamiltonian H_(C)=Σ_(α=1)^(t)h_(α)P_(α) by mapping each binary variable z_(i) to a quantum spinσ_(i) ^(z) and the constraints to the couplings among the quantum spinsσ_(i) ^(z), where P_(α) is a Pauli string (also referred to as a Pauliterm) P_(α)=σ₁ ^(α) ¹ ⊗σ₂ ^(α) ² ⊗ . . . σ_(N) ^(α) ^(N) and σ_(i) ^(α)^(i) is either the identity operator I or the Pauli matrix σ_(i) ^(X),σ_(i) ^(Y), or σ_(i) ^(Z). Here t stands for the number of couplingsamong the quantum spins and h_(α) (α=1, 2, . . . , t) stands for thestrength of the coupling α.

The quantum processor 106 has N qubits and each quantum spin σ_(i) ^(z)(i=1, 2, . . . , N) is encoded in qubit i (i=1, 2, . . . , N) in thequantum processor 106. For example, the spin-up and spin-down states ofthe quantum spin σ_(i) ^(z) are encoded as |0

and |1

of the qubit i.

In block 804, following the mapping of the selected combinatorialoptimization problem onto a model Hamiltonian H_(C)=Σ_(α=1)^(t)h_(α)P_(α), a set of variational parameters ({right arrow over(θ)}=θ₁, θ₂, . . . , θ_(N) for the VQE algorithm, ({right arrow over(γ)}=γ₁, γ₂, . . . , γ_(p), {right arrow over (β)}=β₁, β₂, . . . ,β_(p)) for the QAOA) is selected, by the classical computer 102, toconstruct a sequence of gates (also referred to a “trial statepreparation circuit”) A({right arrow over (θ)}) for the VQE or A({rightarrow over (γ)}, {right arrow over (β)}) for the QAOA, which preparesthe quantum processor 106 in a trial state |Ψ({right arrow over (θ)})

for the VQE or |Ψ({right arrow over (γ)}, {right arrow over (β)})

for the QAOA. For the initial iteration, a set of variational parameters{right arrow over (θ)} in the VQE may be chosen randomly. In the QAOA, aset of variational parameters ({right arrow over (γ)}, {right arrow over(β)}) may be randomly chosen for the initial iteration.

This trial state |Ψ({right arrow over (θ)})

, |Ψ({right arrow over (γ)}, {right arrow over (β)})

is used to provide an expectation value of the model Hamiltonian H_(C).

In the VQE algorithm, the trial state preparation circuit A({right arrowover (θ)}) may be constructed by known methods, such as the unitarycoupled cluster method, based on the model Hamiltonian H_(C) and theselected set of variational parameters {right arrow over (θ)}.

In the QAOA, the trial state preparation circuit A({right arrow over(γ)}, {right arrow over (β)}) includes p layers (i.e., p-timerepetitions) of an entangling circuit U(γ_(l)) that relates to the modelHamiltonian H_(C) (U(γ_(l))=e^(−iγ) ^(l) ^(H) ^(C) ) and a mixingcircuit U_(Mix)(β_(l)) that relates to a mixing term H_(B)=Σ_(i=1)^(n)σ_(i) ^(X) (U_(Mix)(β_(l))=e^(−iβ) ^(l) ^(H) ^(B) ) (l=1, 2, . . . ,p) asA({right arrow over (γ)},{right arrow over (β)})=U _(Mix)(β_(p))U_(Mix)(β_(p-1))U(γ_(p-1)) . . . U _(Mix)(β₁)U(γ₁).Each term σ_(i) ^(X) in the mixing term H_(B) corresponds to a π/2-pulse(as described above in relation to FIG. 4) applied to qubit i in thequantum processor 106.

To allow the application of the trial state preparation circuit A({rightarrow over (θ)}), A({right arrow over (γ)}, {right arrow over (β)}) on aNISQ device, the number of the quantum gate operations need to be small(i.e., shallow circuits) such that errors due to the noise in the NISQdevice are not accumulated. However, as the problem size increases, thecomplexity of the trial state preparation circuit A({right arrow over(θ)}), A({right arrow over (γ)}, {right arrow over (β)}) may increaserapidly, leading to deep circuits (i.e., an increased number of timesteps required to execute gate operations in circuits to construct)required to construct the trial state preparation circuit A({right arrowover (θ)}), A({right arrow over (γ)}, {right arrow over (β)}).Furthermore, some trial state preparation circuit A({right arrow over(θ)}), A({right arrow over (γ)}, {right arrow over (β)}) that aredesigned hardware-efficiently with shallow circuits (i.e., a decreasednumber of time steps required to execute gate operations) may notprovide a large enough variational search space to find the lowestenergy of the model Hamiltonian H_(C).

In the embodiments described herein, the terms in the model HamiltonianH_(C) are grouped into sub-Hamiltonians H_(λ) (λ=1, 2, . . . , u), whereu is the number of sub-Hamiltonians (i.e., H_(C)=Σ_(λ=1) ^(u)H_(λ)), andthe trial state preparation circuit A({right arrow over (θ)}), A({rightarrow over (γ)}, {right arrow over (β)}) is replaced with a reducedstate preparation circuit A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC)^(λ)({right arrow over (γ)}, {right arrow over (β)}) to evaluate anexpectation value of each sub-Hamiltonian H_(λ). The reduced statepreparation circuit A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC)^(λ)({right arrow over (γ)}, {right arrow over (β)}) for asub-Hamiltonian H_(λ) is constructed by a set of gate operations thatcan influence an expectation value of the sub-Hamiltonian H_(λ)(referred to as the past causal cone (PCC) of the sub-Hamiltonian).Other gate operations (that do not influence the expectation value ofthe sub-Hamiltonian H_(λ)) in the trial state preparation circuitA({right arrow over (θ)}), A({right arrow over (γ)}, {right arrow over(β)}) are removed in the reduced state preparation circuits A_(PCC)^(λ)({right arrow over (θ)}), A_(PCC) ^(λ)({right arrow over (γ)},{right arrow over (β)}). In some embodiments, sub-Hamiltonians H_(λ) ofthe model Hamiltonian H_(C) may respectively correspond to Pauli termsP_(α) in the model Hamiltonian H_(C). In some embodiments, asub-Hamiltonian H_(λ) is a collection of more than one Pauli terms P_(α)in the model Hamiltonian H_(C).

For example, the model Hamiltonian H_(C)=σ₁ ^(Z)⊗σ₂ ^(Z)+σ₂ ^(Z)⊗σ₃^(Z)+σ₃ ^(Z)⊗σ₄ ^(Z)+σ₁ ^(Z)⊗σ₄ ^(Z) defined on a system of four qubits(qubit 1, 2, . . . , 4) may be grouped into four sub-Hamiltonians, H₁=σ₁^(Z)⊗σ₂ ^(Z), H₂=σ₂ ^(Z)⊗σ₃ ^(Z), H₃=σ₃ ^(Z)⊗σ₄ ^(Z), and H₄=σ₁ ^(Z)⊗σ₄^(Z). FIG. 9A illustrates the trial state preparation circuit (A({rightarrow over (γ)}, {right arrow over (β)})=U(γ₁)U_(Mix)(β₁)) 900, wherep=1. The mixing circuit U_(Mix)(β_(l)) can be implemented bysingle-qubit rotation gates 902, 904, 906, 908 on qubits 1, 2, 3, and 4,respectively. The entangling circuit U(γ₁) is related to the modelHamiltonian H_(C) as described above. The first term σ₁ ^(Z)⊗σ₂ ^(Z) inthe model Hamiltonian H_(C) can be implemented in combination ofcontrolled-NOT gates 910 on qubit 2 conditioned on qubit 1 and targetedon qubit 2, and a single-qubit rotation gate 912 on qubit 2 about theZ-axis of the Bloch sphere 400 by a polar angle γ₁/2. As one willappreciate, the implementation of such gates can be performed bycombining properly adjusted XX-gate operation between qubits 1 and 2 andcomposite pulses applied to qubits 1 and 2. Other terms in the modelHamiltonian H_(C) can be implemented similarly. FIG. 9B illustrates thereduced state preparation circuits (A_(PCC) ¹({right arrow over (γ)},{right arrow over (β)})) 914 to evaluate an expectation value of thesub-Hamiltonian H₁=σ₁ ^(Z)⊗σ₂ ^(Z). Since qubits 3 and 4 do not affectthe expectation value of the sub-Hamiltonian H₁=σ₁ ^(Z)⊗σ₂ ^(Z), the setof gates 916 and the single-qubit rotation gates 906, 908 (illustratedin FIG. 9A) that are applied only to qubits 3 and 4 in the trial statepreparation circuit A({right arrow over (γ)}, {right arrow over (β)})are removed. FIG. 9C illustrates the reduced state preparation circuits(A_(PCC) ²({right arrow over (γ)}, {right arrow over (β)})) 918 toevaluate an expectation value of the sub-Hamiltonian H₂=σ₂ ^(Z)⊗σ₃ ^(Z).Since qubits 1 and 4 do not affect the expectation value of thesub-Hamiltonian H₂=σ₂ ^(Z)⊗σ₃ ^(Z), the set of gates 920 and thesingle-qubit rotation gates 902, 908 (illustrated in FIG. 9A) that areapplied only to qubits 1 and 4 in the trial state preparation circuitA({right arrow over (γ)}, {right arrow over (β)}) are removed. Thereduced state preparation circuits A_(PCC) ^(λ)({right arrow over (γ)},{right arrow over (β)}) to evaluate an expectation value of othersub-Hamiltonians H_(λ) can be constructed similarly.

With the reduced trial state preparation circuit A_(PCC) ^(λ)({rightarrow over (θ)}), A_(PCC) ^(λ)({right arrow over (γ)}, {right arrow over(β)}) for a sub-Hamiltonian H_(λ), a trial state |Ψ_(λ)({right arrowover (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

is prepared on the quantum processor 106 to evaluate an expectation ofthe sub-Hamiltonian H_(λ). This step is repeated for all of thesub-Hamiltonians H_(λ) (λ=1, 2, . . . , u). The expectation value of themodel Hamiltonian H_(C) is a sum of the expectation values of all of thesub-Hamiltonians H_(λ) (λ=1, 2, . . . , u). The use of the reduced trialstate preparation circuit A_(PCC) ^(λ)({right arrow over (θ)}), A_(PCC)^(λ)({right arrow over (γ)}, {right arrow over (β)}) reduces the numberof gate operations to apply on the quantum processor 106. Thus, a trialstate |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

can be constructed without accumulating errors due to the noise in theNISQ device.

In block 806, following the selection of a set of variational parameters{right arrow over (θ)}, ({right arrow over (γ)}, {right arrow over(β)}), the quantum processor 106 is set in an initial state |Ψ₀

by the system controller 104. In the VQE algorithm, the initial state|Ψ₀

may correspond to an approximate ground state of the system that iscalculated by a classical computer or an approximate ground state thatis empirically known to one in the art. In the QAOA algorithm, theinitial state |Ψ₀

may be in the hyperfine ground state of the quantum processor 106 (inwhich all qubits are in the uniform superposition over computationalbasis states (in which all qubits are in the superposition of |0

and |1

, |0

+|1

). A qubit can be set in the hyperfine ground state |0

by optical pumping and in the superposition state |0

+|1

by application of a proper combination of single-qubit operations(denoted by “H” in FIG. 7) to the hyperfine ground state |0

.

In block 808, following the preparation of the quantum processor 106 inthe initial state |Ψ₀

, the trial state preparation circuit A({right arrow over (θ)}),A({right arrow over (γ)}, {right arrow over (β)}) is applied to thequantum processor 106, by the system controller 104, to construct thetrial state |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

for evaluating an expectation of the sub-Hamiltonian H_(λ). The reducedtrial state preparation circuit A_(PCC) ^(λ)({right arrow over (θ)}),A_(PCC) ^(λ)({right arrow over (γ)}, {right arrow over (β)}) isdecomposed into series of XX-gate operations (XX gates) and single-qubitoperations (R gates) and optimized by the classical computer 102. Theseries of XX-gate operations (XX gates) and single-qubit operations (Rgates) can be implemented by application of a series of laser pulses,intensities, durations, and detuning of which are appropriately adjustedby the classical computer 102 on the set initial state |Ψ₀

and transform the quantum processor from the initial state |Ψ₀

to trial state |Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

.

In block 810, following the construction of the trial state|Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

on the quantum processor 106, the expectation value F_(λ)({right arrowover (θ)})=

Ψ_(λ)({right arrow over (θ)})|H_(λ)|Ψ_(λ)({right arrow over (θ)})

F_(λ)({right arrow over (γ)}, {right arrow over (β)})=

Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})|H_(λ)|Ψ_(λ)({rightarrow over (γ)}, {right arrow over (β)})

of the sub-Hamiltonian H_(λ))λ=1, 2, . . . , u) is measured by thesystem controller 104. Repeated measurements of populations of thetrapped ions in the group 106 of trapped ions the trial state|Ψ_(λ)({right arrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

(by collecting fluorescence from each trapped ion and mapping onto thePMT 110) yield the expectation value the sub-Hamiltonian H_(λ).

In block 812, following the measurement of the expectation value of thesub-Hamiltonian H_(λ) (λ=1, 2, . . . , u), blocks 806 to 810 for anothersub-Hamiltonian H_(λ) (λ=1, 2, . . . , u) until the expectation valuesof all the sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u) in the modelHamiltonian H_(C)=Σ_(λ=1) ^(u)H_(λ) have been measured by the systemcontroller 104.

In block 814, following the measurement of the expectation values of allthe sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u), a sum of the measuredexpectation values of all the sub-Hamiltonian H_(λ) (λ=1, 2, . . . , u)of the model Hamiltonian H_(C)=Σ_(λ=1) ^(u)H_(λ) (that is, the measuredexpectation value of the model Hamiltonian H_(C), F({right arrow over(θ)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (θ)}), F({right arrow over(γ)}, {right arrow over (β)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (γ)},{right arrow over (β)}) is computed, by the classical computer 102.

In block 816, following the computation of the measured expectationvalue of the model Hamiltonian H_(C), the measured expectation valueF({right arrow over (γ)}, {right arrow over (β)}) of the modelHamiltonian H_(C) is compared to the measured expectation value of themodel Hamiltonian H_(C) in the previous iteration, by the classicalcomputer 102. If a difference between the two values is less than apredetermined value (i.e., the expectation value sufficiently convergestowards a fixed value), the method proceeds to block 820. If thedifference between the two values is more than the predetermined value,the method proceeds to block 818.

In block 818, another set of variational parameters {right arrow over(θ)}, ({right arrow over (γ)}, {right arrow over (β)}) for a nextiteration of blocks 806 to 816 is computed by the classical computer102, in search for an optimal set of variational parameters {right arrowover (θ)}, ({right arrow over (γ)}, {right arrow over (β)}) to minimizethe expectation value of the model Hamiltonian H_(C), F({right arrowover (θ)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over (θ)}), F({right arrowover (γ)}, {right arrow over (β)})=Σ_(λ=1) ^(u)F_(λ)({right arrow over(γ)}, {right arrow over (β)}). That is, the classical computer 102 willexecute a classical optimization method to find the optimal set ofvariational parameters {right arrow over (θ)}, ({right arrow over (γ)},{right arrow over (β)})

$\left( {{\min\limits_{\theta}\;\underset{\overset{\rightarrow}{\gamma},\overset{\rightarrow}{\beta}}{F\left( \overset{\rightarrow}{\theta} \right)}},{\min\mspace{11mu}{F\left( {\overset{\rightarrow}{\gamma},\overset{\rightarrow}{\beta}} \right)}}} \right).$Example of conventional classical optimization methods includesimultaneous perturbation stochastic approximation (SPSA), particleswarm optimization (PSO), Bayesian optimization (BO), and Nelder-Mead(NM).

In block 820, the classical computer 102 will typically output theresults of the variational search to a user interface of the classicalcomputer 102 and/or save the results of the variational search in thememory of the classical computer 102. The results of the variationalsearch will include the measured expectation value of the modelHamiltonian H_(C) in the final iteration corresponding to the minimizedenergy of the system in the selected quantum chemistry problem, or theminimized value of the objective function C(z)=Σ_(α=1) ^(t)h_(α)C_(α)(z) of the selected combinatorial optimization problem (e.g., a shortestdistance for all of the trips visiting all given cities in a travellingsalesman problem) and the measurement of the trail state |Ψ_(λ)({rightarrow over (θ)})

, |Ψ_(λ)({right arrow over (γ)}, {right arrow over (β)})

in the final iteration corresponding to the configuration of electronsor spins that provides the lowest energy of the system, or the solutionto the N-bit string (z=z₁ z₂ . . . z_(N)) that provides the minimizedvalue of the objective function C(z)=Σ_(α=1) ^(t)h_(α)C_(α)(z) of theselected combinatorial optimization problem (e.g., a route of the tripsto visit all of the given cities that provides the shortest distance fora travelling salesman).

The variational search reduced trial state preparation circuitsdescribed herein provides an improved method for obtaining a solution toan optimization problem by the Variational Quantum Eigensolver (VQE)algorithm or the Quantum Approximate Optimization Algorithm (QAOA) on ahybrid quantum-classical computing system. Thus, the feasibility that ahybrid quantum-classical computing system may allow solving problems,which are not practically feasible on classical computers, or suggest aconsiderable speed up with respect to the best known classical algorithmeven with a noisy intermediate-scale quantum device (NISQ) device.

While the foregoing is directed to specific embodiments, other andfurther embodiments may be devised without departing from the basicscope thereof, and the scope thereof is determined by the claims thatfollow.

The invention claimed is:
 1. A method of performing computation in ahybrid quantum-classical computing system comprising a classicalcomputer and a quantum processor, comprising: computing, by a classicalcomputer, a model Hamiltonian onto which a selected problem is mapped,wherein the model Hamiltonian comprises a plurality of sub-Hamiltonians;setting a quantum processor in an initial state, wherein the quantumprocessor comprises a plurality of trapped ions, each of which has twofrequency-separated states defining a qubit; transforming the quantumprocessor from the initial state to a trial state based on each of theplurality of sub-Hamiltonians and an initial set of variationalparameters by applying a reduced trial state preparation circuit to thequantum processor; measuring an expectation value of each of theplurality of sub-Hamiltonians on the quantum processor; and determining,by the classical computer, if a difference between the measuredexpectation value of the model Hamiltonian is more or less than apredetermined value, wherein the classical computer either: selectsanother set of variational parameters based on a classical optimizationmethod if it is determined that the difference is more than thepredetermined value and then: sets the quantum processor in the initialstate, transforms the quantum processor from the initial state to a newtrial state based on each of the plurality of sub-Hamiltonians and theanother set of variational parameters by applying a new reduced trialstate preparation circuit to the quantum processor, and measures anexpectation value of the each of the plurality of sub-Hamiltonians onthe quantum processor after transforming the quantum processor to thenew trial state; or outputs the measured expectation value of the modelHamiltonian as an optimized solution to the selected problem if it isdetermined that the difference is less than the predetermined value. 2.The method according to claim 1, wherein the reduced trial statepreparation circuit does not include gate operations that do notinfluence the expectation value of the each of the plurality ofsub-Hamiltonians.
 3. The method according to claim 1, wherein if it isdetermined that the difference is more than the predetermined value, thedetermining step is repeated.
 4. The method according to claim 1,wherein the problem to be solved is finding a lowest energy of amany-particle quantum system.
 5. The method according to claim 4,further comprising: selecting, by the classical computer, the initialset of variational parameters, wherein the initial set of variationalparameters is selected randomly.
 6. The method according to claim 4,wherein setting the quantum processor in the initial state comprisingsetting the plurality of trapped ions in the quantum processor in anapproximate state of the many-particle quantum system that is calculatedby the classical computer.
 7. The method according to claim 1, whereinthe problem to be solved is a combinatorial optimization problem.
 8. Themethod according to claim 7, further comprising: selecting, by theclassical computer, the initial set of variational parameters, whereinthe initial set of variational parameters is selected by the classicalcomputer randomly.
 9. The method according to claim 7, wherein settingthe quantum processor in the initial state comprising setting, by asystem controller, each trapped ion in the quantum processor in asuperposition of the two frequency-separated states.
 10. A hybridquantum-classical computing system, comprising: a quantum processorcomprising a group of trapped ions, each of the trapped ions having twohyperfine states defining a qubit; one or more lasers configured to emita laser beam, which is provided to trapped ions in the quantumprocessor; and a classical computer configured to: select a problem tobe solved; compute a model Hamiltonian onto which the selected problemis mapped, wherein the model Hamiltonian comprises a plurality ofsub-Hamiltonians; select a set of variational parameters; set thequantum processor in an initial state; transform the quantum processorfrom the initial state to a trial state based on each of the pluralityof sub-Hamiltonians and an initial set of variational parameters byapplying a reduced trial state preparation circuit to the quantumprocessor; measure an expectation value of each of the plurality ofsub-Hamiltonians on the quantum processor; and determine if a differencebetween the measured expectation value of the model Hamiltonian is moreor less than a predetermined value, wherein the classical computereither: selects another set of variational parameters based on aclassical optimization method if it is determined that the difference ismore than the predetermined value and then: sets the quantum processorin the initial state, transforms the quantum processor from the initialstate to a new trial state based on each of the plurality ofsub-Hamiltonians and the another set of variational parameters byapplying a new reduced trial state preparation circuit to the quantumprocessor, and measures an expectation value of each of the plurality ofsub-Hamiltonians on the quantum processor after transforming the quantumprocessor to the new trial state; or outputs the measured expectationvalue of the model Hamiltonian as an optimized solution to the selectedproblem if it is determined that the difference is less than thepredetermined value.
 11. The hybrid quantum-classical computing systemaccording to claim 10, wherein the reduced trial state preparationcircuit does not include gate operations that do not influence theexpectation value of the each of the plurality of sub-Hamiltonians. 12.The hybrid quantum-classical computing system according to claim 10,wherein if it is determined that the difference is more than thepredetermined value, the classical computer repeats the determiningstep.
 13. The hybrid quantum-classical computing system according toclaim 10, wherein the problem to be solved is finding a lowest energy ofa many-particle quantum system.
 14. The hybrid quantum-classicalcomputing system according to claim 13, wherein the classical computerfurther selects the initial set of variational parameters randomly. 15.The hybrid quantum-classical computing system according to claim 13,wherein the initial state is an approximate state of the many-particlequantum system that is calculated by the classical computer.
 16. Thehybrid quantum-classical computing system according to claim 10, whereinthe problem to be solved is a combinatorial optimization problem. 17.The hybrid quantum-classical computing system according to claim 16,wherein the classical computer selects the initial set of variationalparameters randomly.
 18. The hybrid quantum-classical computing systemaccording to claim 16, wherein the initial state is a superposition ofthe two hyperfine states.
 19. A hybrid quantum-classical computingsystem comprising non-volatile memory having a number of instructionsstored therein which, when executed by one or more processors, causesthe hybrid quantum-classical computing system to perform operationscomprising: computing a model Hamiltonian onto which a selected problemis mapped, wherein the model Hamiltonian comprises a plurality ofsub-Hamiltonians; setting a quantum processor in an initial state,wherein the quantum processor comprises a plurality of trapped ions,each of which has two frequency-separated states defining a qubit;transforming the quantum processor from the initial state to a trialstate based on each of the plurality of sub-Hamiltonians and an initialset of variational parameters by applying a reduced trial statepreparation circuit to the quantum processor; measuring an expectationvalue of each of the plurality of sub-Hamiltonians on the quantumprocessor; and determining if a difference between the measuredexpectation value of the model Hamiltonian is more or less than apredetermined value, wherein the instructions further cause the hybridquantum-classical computing system to either: select another set ofvariational parameters based on a classical optimization method if it isdetermined that the difference is more than the predetermined value andthen: set the quantum processor in the initial state, transform thequantum processor from the initial state to a new trial state based oneach of the plurality of sub-Hamiltonians and the another set ofvariational parameters by applying a new reduced trial state preparationcircuit to the quantum processor, and measure an expectation value ofthe each of the plurality of sub-Hamiltonians on the quantum processorafter transforming the quantum processor to the new trial state; oroutput the measured expectation value of the model Hamiltonian as anoptimized solution to the selected problem if it is determined that thedifference is less than the predetermined value.
 20. The hybridquantum-classical computing system according to claim 19, wherein thereduced trial state preparation circuit does not include gate operationsthat do not influence the expectation value of the each of the pluralityof sub-Hamiltonians.